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Y. Sumino (Tohoku Univ.)

Basics of potential-NRQCD and Quarkonium Spectroscopy. Y. Sumino (Tohoku Univ.). Heavy Quarkonium : . ?. For , bound-state theory based on pert. QCD is valid. gluons with decouples. Appropriate EFT. NRQCD potential NRQCD ( pNRQCD )

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Y. Sumino (Tohoku Univ.)

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  1. Basics of potential-NRQCD and Quarkonium Spectroscopy Y. Sumino (Tohoku Univ.)

  2. Heavy Quarkonium: ? For , bound-state theory based on pert. QCD is valid. gluons with decouples Appropriate EFT NRQCD potential NRQCD (pNRQCD) velocity NRQCD (vNRQCD) Caswell, Lepage Pineda, Soto Manohar, Stewart

  3. Virtue of EFT Principle sym. & small parameter • Constrain possible interactions • Systematic expansion • Matching to full theory • OPE (separating pert.vsnon-pert. effects) cf. Chiral PT Wilson coeffs. Matrix elements of operators

  4. Relevant d.o.f. and interacting via potential(s) + IR gluons (ultrasoft gluons) To undestand systematic expansion, consider hydrogen-like atom (pNRQED) Bound-state IR photon Bohr radius All other (short-dist.) modes are “integrated out” Effects incorporated into Wilson coefficients=“potentials”. Expansion parameters:

  5. Lagrangian where should be expanded in . Field redefinition Expansion parameters: gauge singlet

  6. … Lagrangian gauge inv. at each order of -expansion dipole interaction LO Lagrangian : electric field at e.g. Lamb shift US photon Propagator of hydrogen-like atom in QM: with

  7. Systematic expansion in “Potential”=Wilson coeff.: function of Non-Local in Local in Integrating out

  8. Potential-NRQCD and OPE of QCD potential QCD potential: : as OPE: expansion in Brambilla, Pineda, Soto, Vairo US gluon singlet octet singlet UV contr. IR contr.

  9. OPE: expansion in Brambilla, Pineda, Soto, Vairo US gluon singlet octet singlet UV contr. IR contr.

  10. non-pert. matrix element Potential NRQCD is valid below cut-off scale , where . • : Wilson coeff., , perturbative • : non-pert. matrix element + OPE: expansion in Brambilla, Pineda, Soto, Vairo US gluon prediction of OPE and pert. QCD singlet octet singlet UV contr. IR contr. Folklore: pert. , non-pert. Wrong at r < 0.5 fm !

  11. Y.S. Accuracy of perturbative prediction of improved drastically around 1998, due to discovery of renormalon cancellation in . Hoang, Smith, Steltzer, Willenbrock; Beneke

  12. Application to quarkonium spectroscopy and determination of . • Global level structure of bottomonium is reproduced. Brambilla, Sumino, Vairo • Determination of bottom and charm quark MS masses: • Brambilla, Sumino, Vairo • Fine and hyperfine splittings of charmonium/bottomonium reproduced. Recksiegel, Y.S In particular, mass of is predicted correctly. Recksiegel, Y.S. However, mass of disagrees: (prediction) (exp.09) • Relation between lattice and MS is accurately measured (quenched approx.) • Y.S.

  13. D.o.f. that are integrated out in pNRQCD: hard ( , + gluon): soft ( , + gluon): potential (gluon): Dynamical d.o.f. in pNRQCD: potential ( , ): ultrasoft (gluon): No other scales are known to play roles in Coulombic bound-states. Beneke, Smirnov

  14. § 3. QCD Potential: OPE Analysis at

  15. QCD Potential: Pert. QCD

  16. Lattice QCD: Determination of

  17. OPE of non-pert. contr. Wilson coeff. cancel

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